"""
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
What is the 10001st prime number?
"""

import time
from math import log


def timer(func):
    def wrapper(*args, **kwargs):
        start = time.time()
        result = func(*args, **kwargs)
        end = time.time()
        print(f"{func.__name__} took {end - start:.6f} seconds")
        return result

    return wrapper


@timer
def sieve_nth_prime(n: int) -> int | None:
    if n == 1:
        return 2
    # 估算上限：第n个质数约在 n*log(n) 附近
    limit = int(n * (log(n) + log(log(n)))) + 10 if n > 6 else 20
    sieve = [True] * limit
    sieve[0:2] = [False, False]  # 0和1不是质数

    count = 0
    for p in range(2, limit):
        if sieve[p]:
            count += 1
            if count == n:
                return p
            # 标记倍数
            sieve[p * p : limit : p] = [False] * ((limit - 1 - p * p) // p + 1)

    return None


if __name__ == "__main__":
    print(sieve_nth_prime(10001))